- Draw a vertical, dotted "wall" at the possible point of discontinuity.
- Determine which piece(s) of your function will have a closed circle at your wall and which one(s) will have an open circle.
- Determine which function you'll use for all your x's to the left of the wall and which function you'll use for the right.
- Graph the top function (use transformations); erase everything to the left or the right of your wall, depending on your decision from Step 3.
- Repeat Step 4 for the bottom function. Erase the oppose piece this time.

The key, for me, is "the wall." I've used this concept before in analyzing limits in calculus graphically, but I don't know why it didn't dawn on me to use the same concept here until recently. It worked like a charm--hardly any students drew the nonsensical, non-function relations that I've seen in the past. Also, hopefully this gives us a leg up when we get to limits next semester. Fingers crossed!

What a great idea! I, too, have wrestled with this skill for years.

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